University of Southern California and University of Melbourne We introduce a new family of distributions to approximate P(W ∈A) for A ⊂ {. . . ,−2,−1,0,1,2, . . .} and W a sum of independent integer-valued random variables ξ1, ξ2, . . . , ξn with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem s...

In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and t...

In this paper, we obtain an explicit Berry-Esseen bound in the central limit theorem for nonlinear statistics with bounded differences. Some examples are provided as well.

Bounds in the mean central limit theorem, where the L1 distance is used to measure the discrepancy of the distribution Fn of a standardized sum of i.i.d. random variables with distributionG from the normal, is of some interest when the normal approximation is to be applied over some wide, and perhaps unspecified range of values. Esseen (1958) showed that the limiting value lim n→∞ n||Fn − Φ||1 ...

Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from these new bounds vanishes at a rate O(n ); provided that the common distribution of the summands po...

For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn = Zn/E[Zn|ξ] to its limitW∞: we show a central limit theorem forW∞−Wn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distrib...

To lower bound the effective dimension for local Hamiltonian models, we will use a theorem from [1, 2] as a stepping stone. This is a quantum version of the BerryEsseen theorem. The Berry-Esseen theorem is a more powerful statement than the central limit theorem, as it gives the rate of convergence of a distribution to a Gaussian. Let H = ∑ ν Eν |ν〉〈ν| be k-local, i.e., let H be of the form H =...